565 research outputs found
Sufficient conditions for the existence of Zeno behavior in a class of nonlinear hybrid systems via constant approximations
The existence of Zeno behavior in hybrid systems
is related to a certain type of equilibria, termed Zeno equilibria,
that are invariant under the discrete, but not the continuous,
dynamics of a hybrid system. In analogy to the standard
procedure of linearizing a vector field at an equilibrium point to
determine its stability, in this paper we study the local behavior
of a hybrid system near a Zeno equilibrium point by considering
the value of the vector field on each domain at this point, i.e., we
consider constant approximations of nonlinear hybrid systems.
By means of these constant approximations, we are able to
derive conditions that simultaneously imply both the existence
of Zeno behavior and the local exponential stability of a Zeno
equilibrium point. Moreover, since these conditions are in terms
of the value of the vector field on each domain at a point, they
are remarkably easy to verify
Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems
This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts. This class of systems naturally display Zeno behavior-an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint
Stability of Zeno Equilibria in Lagrangian Hybrid Systems
This paper presents both necessary and sufficient
conditions for the stability of Zeno equilibria in Lagrangian hybrid systems, i.e., hybrid systems modeling mechanical systems undergoing impacts. These conditions for stability are motivated by the sufficient conditions for Zeno behavior in Lagrangian hybrid systems obtained in [11]—we show that the same conditions that imply the existence of Zeno behavior near Zeno equilibria imply the stability of the Zeno equilibria. This paper, therefore, not only presents conditions for the stability of Zeno equilibria, but directly relates the stability of Zeno equilibria to the existence of Zeno behavior
Formal and practical completion of Lagrangian hybrid systems
This paper presents a method for completing
Lagrangian hybrid systems models in a formal manner. That
is, given a Lagrangian hybrid system, i.e., a hybrid system that
models a mechanical system undergoing impacts, we present
a systematic method in which to extend executions of this
system past Zeno points by adding an additional domain to
the hybrid model. Moreover, by utilizing results that provide
sufficient conditions for Zeno behavior and for stability of Zeno
equilibria in Lagrangian hybrid systems, we are able to give
explicit bounds on the error incurred through the practical
simulation of these completed hybrid system models. These
ideas are illustrated on a series of examples, and are shown
to be consistent with observed reality
A Sum-of-Squares Approach to the Analysis of Zeno Stability in Polynomial Hybrid Systems
Hybrid dynamical systems can exhibit many unique phenomena, such as Zeno
behavior. Zeno behavior is the occurrence of infinite discrete transitions in
finite time. Zeno behavior has been likened to a form of finite-time asymptotic
stability, and corresponding Lyapunov theorems have been developed. In this
paper, we propose a method to construct Lyapunov functions to prove Zeno
stability of compact sets in cyclic hybrid systems with parametric
uncertainties in the vector fields, domains and guard sets, and reset maps
utilizing sum-of-squares programming. This technique can easily be applied to
cyclic hybrid systems without parametric uncertainties as well. Examples
illustrating the use of the proposed technique are also provided
Stability and completion of Zeno equilibria in Lagrangian hybrid systems.
Abstract This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts. This class of systems naturally display Zeno behavior-an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint
Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems
This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts. This class of systems naturally display Zeno behavior-an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint
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